Gamut mapping is a method used to modify a representation of a color image to fit into a constrained color space of a given rendering medium. A laser-jet color printer that attempts to reproduce a color image on regular paper would have to map the photographed picture colors in a given color range, also known as the image “color gamut,” into the given printer/page color gamut. Conventional gamut mapping methods involve a pixel by pixel mapping (usually a pre-defined look-up table) and ignore the spatial color configuration. More recently, spatial dependent approaches were proposed for gamut mapping. However, these solutions are either based on heurestic assumptions or involve a high computational cost. An example of such a method is discussed in “Color Gamut Mapping Based on a Perceptual Image Difference Measure,” S. Nakauchi, et al., Color Research and Application, Vol. 24, pp. 280-291 (1999).
Another method relies on preserving the magnitude of gradients in an original image, while projecting the gradients onto the target gamut as a constraint. This multi-scale property is achieved by sampling the image around each pixel with exponentially increasing sampling intervals while the sampling is done along vertical and horizontal axes. The method involves modulating an L2 measure for image difference by human contrast sensitivity functions. The method uses a model in which the contrast sensitivity function is a linear combination of three spatial band-pass filters H1, H2, H3, given in the spatial-frequency domain (or h1, h2, h3, as their corresponding spatial filters), as shown in FIG. 1.
For gamut mapping of the image u0 in the CIELAB (which stands for Commission International de I'Eclairage (CIE) with L, A, and B (more commonly L, a, b) respectively representing converted signals for cyan (C), magenta (M) and yellow (Y)) space, the method minimizes the functional                                           E            ⁡                          (                                                u                  L                                ,                                  u                  a                                ,                                  u                  b                                            )                                =                                    ∑                              i                =                1                            3                        ⁢                                                   ⁢                                          ∑                                  c                  ∈                                      {                                          L                      ,                      a                      ,                      b                                        }                                                              ⁢                                                ∫                  Ω                                ⁢                                                                            (                                                                        h                          i                          c                                                *                                                  (                                                                                    u                              c                                                        -                                                          u                              0                              c                                                                                )                                                                    )                                        2                                    ⁢                                      ⅆ                    Ω                                                                                      ,                            (        1        )            subject to {uL, ue, ub}ε .In Equation (1), h1c is the filter corresponding to the spectral channel c ε {L, a, b} the i ε {1, 2, 3} ‘contrast sensitivity’ mod, Ω is the image domain, and  is the target gamut. Note that a total of nine filters H1c are involved, three for each spectral channel and a total of three spectral channels.
The filters Hc, are modeled by Gaussians shifted in the spatial frequency domain. H1c is a special case where the shift is zero. The spread function h1c corresponding to H1c is the Inverse Fourier Transform of H1c, and is thus a Gaussian.
As depicted in the first line of FIG. 2 for the case of H2c and H3c, spatial frequency shifts are a convolution with correspondingly shifted delta functions in the spatial frequency domain. The soread functions h1c, (for i=2, 3) depicted in line 2 of FIG. 2 are the Inverse Fourier Transforms of the corresponding H1c. As such, they are multiplications of the Inverse Fourier Transforms of the Gaussian and the corresponding shifted delta functions, namely a multiplication of a Gaussian and a harmonic function. As depicted graphically in line 3 of FIG. 2, harmonically modulated Gaussians with different harmonic periods correspond to derivatives at different scales. Thus, the gradient approximation filters H2c and H3c may be denoted by ∇hu cσ1, and ∇σ2. Note that any band pass filter can be considered as a version of a derivative operator. Furthermore, one possible extension of the 1D derivative to 2D is the gradient. Thus, the minimization of the Equation (1) functional is similar to minimizing the following functional for each channel separately.                               ∫          Ω                ⁢                  |                                    h              1              c                        *                          (                              u                -                                  u                  0                                            )                                ⁢                      |            2                    ⁢                      +                          |                                                ∇                  σ1                  c                                ⁢                                  (                                      u                    -                                          u                      0                                                        )                                            ⁢                              |                2                            ⁢                              +                                  |                                                            ∇                      σ2                      c                                        ⁢                                          (                                              u                        -                                                  u                          0                                                                    )                                                        ⁢                                      |                    2                                    ⁢                                      ⅆ                    Ω                                                                                                          (        2        )            
Roughly speaking, the first term corresponds to the S-CIELAB perceptual measure, while the next two terms capture the need for matching the image variations at two selected scales that were determined by human perception models. One technical difficulty of the spatial filters corresponding to Equation (1) is their large numerical support.
Another simple spatial-spectral measure for human color perception was proposed in “A Spatial Extension of CIELAB for Digital Color Image Reproduction,” Zhang et al., Proceedings of the SID Symposium, Vol. 27, pp. 731-34, (1996). The ‘S-CIELAB’ defines a spatial-spectral measure for human color perception by a composition of spatial band-pass linear filters in the opponent color space followed by the CIELAB Euclidean perceptual color.